Question 1

Find a rectangular equation for the plane curve defined by the parametric equations.
-A baseball pitcher throws a baseball with an initial speed of 140 feet per second at an angle of 20° to the horizontal. The ball leaves the pitcher's hand at a height of 5 feet. Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the ball at its maximum height?
What is the maximum height of the ball? 
A) $x = 124.04 t$ and $y = - 16 t ^ { 2 } + 45.14 t + 5$
$5.856 \mathrm { sec } , 1.411 \mathrm { sec }$,
$31.838$ feet

B) $x = 124.04 t$ and $y = - 16 t ^ { 2 } + 45.14 t + 5$
$5.412 \mathrm { sec } , 1.411 \mathrm { sec }$,
$259.702$ feet

C) $x = 124.04 t$ and $y = - 16 t ^ { 2 } + 45.14 t + 5$
$2.706 \mathrm { sec } , 1.411 \mathrm { sec }$,
$4.99$ feet

D) $x = 131.56 \mathrm { t }$ and $\mathrm { y } = - 16 \mathrm { t } ^ { 2 } + 47.88 \mathrm { t } + 5$
$3.094 \mathrm { sec } , 1.496 \mathrm { sec }$,
$40.82$ feet

Accepted Answer

The answer of Find a rectangular equation for the plane...

Question 2

Find a rectangular equation for the plane curve defined by the parametric equations.
-$$x = 7 \sin t , y = 7 \cos t ; 0 \leq t \leq 2 \pi$$
A) $x ^ { 2 } + y ^ { 2 } = 49$; for $x$ in $- 7 \leq x \leq 7$
B) $y = x ^ { 2 } - 9$; for $x$ in $- 2 \leq x \leq 2$
C) $y = \sqrt { a ^ { 2 } - x ^ { 2 } } = 49$; for $x$ in $- \infty < x < \infty$
D) $y ^ { 2 } - x ^ { 2 } = 49$; for $x$ in $- \infty < x < \infty$

Accepted Answer

Squaring both parametric equations and adding them gives $x^2 + y^2 = 49$, which is a circle with radius 7.

Question 3

Find parametric equations for the rectangular equation.
-

Accepted Answer

The answer of Find parametric equations for the rectangular equation.
-...

Question 4

Find a rectangular equation for the plane curve defined by the parametric equations.
-$$x = 2 t - 1 , y = t ^ { 2 } + 4 ; - 4 \leq t \leq 4$$
A) $y = - \frac { 1 } { 2 } x + 30$; for $x$ in $- 6 \leq x \leq 4$
B) $y = \frac { 1 } { 2 } x ^ { 2 } + 1 ;$ for $x$ in $- 6 \leq x \leq 4$
C) $\mathrm { y } = \mathrm { x } ^ { 2 } + 1 ;$ for $\mathrm { x }$ in $- 2 \leq \mathrm { x } \leq 2$
D) $y = \frac { 1 } { 4 } x ^ { 2 } + \frac { 1 } { 2 } x + \frac { 17 } { 4 } ;$ for $x$ in $- 9 \leq x \leq 7$

Accepted Answer

To find a rectangular equation, we eliminate the parameter $t$. From $x = 2t - 1$, we solve for $t$ to get $t = \frac{x + 1}{2}$. Substituting this into $y = t^2 + 4$, we get $y = \left(\frac{x + 1}{2}\right)^2 + 4$, which simplifies to $y = \frac{1}{4}x^2 + \frac{1}{2}x + \frac{17}{4}$. The range for $x$ is found by substituting the limits of $t$ into the equation for $x$, giving $x$ values from $-9$ to $7$.

Question 5

Find parametric equations for the rectangular equation.
-$$y = x ^ { 4 } - 4$$
A) $x = t ^ { 2 } , y = t ^ { 2 } - 4 ; 0 \leq t < \infty$
B) $x = t ^ { 2 } , y = t ^ { 4 } - 4 ; 0 \leq t < \infty$
C) $x = t , y = t ^ { 2 } - 4 ; 0 \leq t < \infty$
D) $x = t , y = t ^ { 4 } - 4 ; 0 \leq t < \infty$

Accepted Answer

The answer of Find parametric equations for the rectangular equation.
-\[y...

Question 6

Graph the curve whose parametric equations are given.
-$$\begin{array} { l } 
C _ { 1 } : x = 7 \sin t , y = 7 - 7 \cos ^ { 2 } t ; \frac { \pi } { 2 } \leq t \leq \frac { 3 \pi } { 2 } \
C _ { 2 } : x = \ln t , y = \ln t ^ { 2 } ; e ^ { - 4 } \leq t \leq e ^ { 3 } \
C _ { 3 } : x = t ^ { 2 } - 8 , y = t - 3 ; - 4 \leq t \leq 4 \
C _ { 4 } : x = t - 5 , y = t + 2 ; - 4 \leq t \leq 7
\end{array}$$

Accepted Answer

The answer of Graph the curve whose parametric equations are...

Question 7

Graph the curve whose parametric equations are given.
-$x=3 \cos t, y=2 \sin t ; 0 \leq t \leq 2 \pi$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the curve whose parametric equations are...

Question 8

Find a rectangular equation for the plane curve defined by the parametric equations.
-$x = 3 \tan t , y = 2 \sec t ; 0 \leq t \leq 2 \pi$
A) $y = x ^ { 2 } - 9$; for $x$ in $- 3 \leq x \leq 3$
B) $y = 3 \sqrt { 1 + \frac { x ^ { 2 } } { 25 } }$; for $x$ in $- \infty < x < \infty$
C) $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 25 } = 1$; for $x$ in $- \infty < x < \infty$
D) $\frac { y ^ { 2 } } { 9 } + \frac { x ^ { 2 } } { 25 } = 1$; for $x$ in $- \infty < x < \infty$

Accepted Answer

The equation of the ellipse is $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 25 } = 1$.

Question 9

Find a rectangular equation for the plane curve defined by the parametric equations.
-$$y = 4 x - 5$$
A) $y = 4 t , 4 x = t + 5 ; 0 \leq t < \infty$
B) $x = \frac { t } { 4 } , y = t - \frac { 5 } { 4 } ; 0 \leq t < \infty$
C) $x = t , y = 4 t - 5 ; 0 \leq t < \infty$
D) $x = t , y = 4 t ^ { 2 } - 5 ; 0 \leq t < \infty$

Accepted Answer

The given parametric equations can be directly matched with choice C, where $x = t$ and $y = 4t - 5$, which is the same as the given equation $y = 4x - 5$ when substituting $x$ for $t$.

Question 10

Solve the problem.
-Find parametric equations for an object that moves along the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 81 } = 1$ with the motion described.
The motion begins at $( 3,0 )$, is counterclockwise, and requires 5 seconds for a complete revolution.
A) $x = 3 \cos \left( \frac { 2 \pi } { 5 } t \right) , y = - 9 \sin \left( \frac { 2 \pi } { 5 } t \right) , 0 \leq t \leq 5$
B) $x = - 4 \cos \left( \frac { 2 \pi } { 3 } t \right) , y = 6 \sin \left( \frac { 2 \pi } { 3 } t \right) , 0 \leq t \leq 3$
C) $x = - 3 \cos \left( \frac { 2 \pi } { 5 } t \right) , y = - 9 \sin \left( \frac { 2 \pi } { 5 } t \right) , 0 \leq t \leq 5$
D) $x = 4 \cos \left( \frac { 2 \pi } { 3 } t \right) , y = 6 \sin \left( \frac { 2 \pi } { 3 } t \right) , 0 \leq t \leq 3$

Accepted Answer

The correct parametric equations must reflect the shape of the ellipse, which has radii 3 and 9 in the x and y directions, respectively. The motion is counterclockwise and starts at (3,0), which matches the cosine function for x starting at its maximum value. The period of 5 seconds for a complete revolution is accounted for by the factor $\frac{2\pi}{5}t$ in both equations.

Question 11

Find a rectangular equation for the plane curve defined by the parametric equations.
-A baseball player hit a baseball with an initial speed of 160 feet per second at an angle of 40° to the horizontal.The ball was hit at a height of 4 feet off the ground. Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the ball at its maximum height? What is the
Distance the ball traveled? 
A) $x = 114.9 t$ and $y = - 16 t ^ { 2 } + 96.45 t + 3$
$6.059 \mathrm { sec } , 3.014 \mathrm { sec }$,
$1171.774$ feet

B) $x = 114.9 t$ and $y = - 16 t ^ { 2 } + 96.45 t + 3$
$5.997 \mathrm { sec } , 3.014 \mathrm { sec }$,
$689.055$ feet

C) $x = 122.56 t$ and $y = - 16 t ^ { 2 } + 102.88 t + 4$
$6.469 \mathrm { sec } , 3.215 \mathrm { sec }$,
$792.841$ feet

D) $x = 122.56 t$ and $y = - 16 t ^ { 2 } + 102.88 t + 4$
$12.937 \mathrm { sec } , 3.215 \mathrm { sec }$,
$1,585.559$ feet

Accepted Answer

The parametric equations for the motion of the ball can be derived using the initial speed and angle of projection. The horizontal component of the velocity is $v_0 \cos(\theta)$ and the vertical component is $v_0 \sin(\theta)$, where $v_0 = 160$ ft/s and $\theta = 40^\circ$. The acceleration due to gravity acts in the vertical direction and is represented by $-16t^2$ (using $g = 32$ ft/s$^2$ and the negative sign indicates downward acceleration). The initial height is added to the vertical equation. The time in the air is found by setting the vertical position equation to zero and solving for $t$. The time at maximum height is found by setting the derivative of the vertical position equation with respect to time to zero. The distance traveled is the horizontal distance covered in the time the ball is in the air.

Question 12

Find a rectangular equation for the plane curve defined by the parametric equations.
-$$x = 5 \cos t , y = - 2 \sin t ; 0 \leq t \leq 2 \pi$$
A) $4 x ^ { 2 } + 25 y ^ { 2 } = 100 ; - 5 \leq x \leq 5$
B) $4 x ^ { 2 } - 25 y ^ { 2 } = 100 ; x \geq 5$
C) $4 x ^ { 2 } + 25 y ^ { 2 } = 1 ; - \frac { 1 } { 5 } \leq x \leq \frac { 1 } { 5 }$
D) $4 x ^ { 2 } - 25 y ^ { 2 } = 1 ; x \geq \frac { 1 } { 2 }$

Accepted Answer

To eliminate the parameter $t$, use the trigonometric identity $\cos^2 t + \sin^2 t = 1$. From the given equations, $x = 5\cos t$ and $y = -2\sin t$, we can solve for $\cos t$ and $\sin t$ as $\cos t = x/5$ and $\sin t = -y/2$. Substituting these into the identity gives $(x/5)^2 + (-y/2)^2 = 1$, which simplifies to $4x^2 + 25y^2 = 100$. The range for $x$ comes from the maximum and minimum values of $5\cos t$, which are $-5$ and $5$, respectively.

Question 13

Find a rectangular equation for the plane curve defined by the parametric equations.
-Ron throws a ball straight up with an initial speed of 70 feet per second from a height of 3 feet. Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the
Ball at its maximum height? What is the maximum height of the ball? 
A) $x = 0$ and $y = - 16 t ^ { 2 } + 40 t + 6$
$5.284 \mathrm { sec } , 1.25 \mathrm { sec }$,
25 feet

B) $x = 0$ and $y = - 16 t ^ { 2 } + 40 t + 6$
$2.34 \mathrm { sec } , 1.25 \mathrm { sec }$,
$5.99$ feet

C) $x = 0$ and $y = - 16 t ^ { 2 } + 40 t + 6$
$4.679 \mathrm { sec } , 1.25 \mathrm { sec }$,
$181.21$ feet

D) $x = 0$ and $y = - 16 t ^ { 2 } + 40 t + 6$
$2.642 \mathrm { sec } , 1.25 \mathrm { sec }$,
31 feet

Accepted Answer

The answer of Find a rectangular equation for the plane...

Question 14

Graph the curve whose parametric equations are given.
-$\mathrm{x}=\mathrm{t}+2, \mathrm{y}=3 \mathrm{t}-1 ; 0 \leq \mathrm{t} \leq 3$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the curve whose parametric equations are...

Question 15

Find a rectangular equation for the plane curve defined by the parametric equations.
-$$x = t ^ { 3 } + 1 , y = t ^ { 3 } - 10 ; - 2 \leq t \leq 2$$
A) $y = x - 11$; for $x$ in $- 7 \leq x \leq 9$
B) $y = - x - 11$; for $x$ in $- 7 \leq x \leq 9$
C) $y = x ^ { 3 }$; for $x$ in $- 3 \leq x \leq 1$
D) $y = - x ^ { 2 }$; for $x$ in $- 4 \leq x \leq 4$

Accepted Answer

Given $x = t^3 + 1$ and $y = t^3 - 10$, we can eliminate the parameter $t$ by observing that both $x$ and $y$ are defined in terms of $t^3$. Subtracting the equation for $y$ from the equation for $x$ gives $x - y = (t^3 + 1) - (t^3 - 10) = 11$, which simplifies to $y = x - 11$. The range for $x$ is found by substituting the limits of $t$ into the equation for $x$, resulting in $x$ values between $-7$ and $9$.

Question 16

Find a rectangular equation for the plane curve defined by the parametric equations.
-$$\mathrm { x } = 3 \mathrm { t } , \mathrm { y } = \mathrm { t } + 1 ; - 2 \leq \mathrm { t } \leq 3$$
A) $y = \frac { 1 } { 3 } x - 1$; for $x$ in $- \infty < x < \infty$
B) $y = x ^ { 2 } + 1$; for $x$ in $- 2 \leq x \leq 2$
C) $y = - 3 x + 1$; for $x$ in $- \infty < x < \infty$
D) $y = \frac { 1 } { 3 } x + 1$; for $x$ in $- 6 \leq x \leq 9$

Accepted Answer

To find a rectangular equation, we eliminate the parameter $t$. From $x = 3t$, we get $t = \frac{x}{3}$. Substituting this into $y = t + 1$, we get $y = \frac{x}{3} + 1$, which matches option D. The domain for $x$ comes from the given range of $t$, where $x = 3t$ ranges from $-6$ to $9$ when $t$ ranges from $-2$ to $3$.

Question 17

Graph the curve whose parametric equations are given.
-$x=2 t^{2}$ , $y=t+2$ : $- \infty < t < \infty$

A)

B)

C)

D)

Accepted Answer

The answer of Graph the curve whose parametric equations are...

Quiz 9: Analytic Geometry

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 7 \sin t , y = 7 \cos t ; 0 \leq t \leq 2 \pi$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 2 t - 1 , y = t ^ { 2 } + 4 ; - 4 \leq t \leq 4$

Find parametric equations for the rectangular equation. - $y = x ^ { 4 } - 4$

Graph the curve whose parametric equations are given. - $x=3 \cos t, y=2 \sin t ; 0 \leq t \leq 2 \pi$ $Graph the curve whose parametric equations are given. - x=3 \cos t, y=2 \sin t ; 0 \leq t \leq 2 \pi A) B) C) D)$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 3 \tan t , y = 2 \sec t ; 0 \leq t \leq 2 \pi$

Find a rectangular equation for the plane curve defined by the parametric equations. - $y = 4 x - 5$

Solve the problem. -Find parametric equations for an object that moves along the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 81 } = 1$ with the motion described. The motion begins at $( 3,0 )$ , is counterclockwise, and requires 5 seconds for a complete revolution.

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 5 \cos t , y = - 2 \sin t ; 0 \leq t \leq 2 \pi$

Graph the curve whose parametric equations are given. - $\mathrm{x}=\mathrm{t}+2, \mathrm{y}=3 \mathrm{t}-1 ; 0 \leq \mathrm{t} \leq 3$ $Graph the curve whose parametric equations are given. - \mathrm{x}=\mathrm{t}+2, \mathrm{y}=3 \mathrm{t}-1 ; 0 \leq \mathrm{t} \leq 3 A) B) C) D)$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = t ^ { 3 } + 1 , y = t ^ { 3 } - 10 ; - 2 \leq t \leq 2$

Find a rectangular equation for the plane curve defined by the parametric equations. - $\mathrm { x } = 3 \mathrm { t } , \mathrm { y } = \mathrm { t } + 1 ; - 2 \leq \mathrm { t } \leq 3$

Graph the curve whose parametric equations are given. - $x=2 t^{2}$ , $y=t+2$ : $- \infty < t < \infty$ $Graph the curve whose parametric equations are given. - x=2 t^{2} , y=t+2 : - \infty < t < \infty A) B) C) D)$

Quiz 9: Analytic Geometry

Find a rectangular equation for the plane curve defined by the parametric equations. - x=7sin⁡t,y=7cos⁡t;0≤t≤2πx = 7 \sin t , y = 7 \cos t ; 0 \leq t \leq 2 \pix=7sint,y=7cost;0≤t≤2π

Find a rectangular equation for the plane curve defined by the parametric equations. - x=2t−1,y=t2+4;−4≤t≤4x = 2 t - 1 , y = t ^ { 2 } + 4 ; - 4 \leq t \leq 4x=2t−1,y=t2+4;−4≤t≤4

Find parametric equations for the rectangular equation. - y=x4−4y = x ^ { 4 } - 4y=x4−4

Graph the curve whose parametric equations are given. - x=3cos⁡t,y=2sin⁡t;0≤t≤2πx=3 \cos t, y=2 \sin t ; 0 \leq t \leq 2 \pix=3cost,y=2sint;0≤t≤2π

Find a rectangular equation for the plane curve defined by the parametric equations. - x=3tan⁡t,y=2sec⁡t;0≤t≤2πx = 3 \tan t , y = 2 \sec t ; 0 \leq t \leq 2 \pix=3tant,y=2sect;0≤t≤2π

Find a rectangular equation for the plane curve defined by the parametric equations. - y=4x−5y = 4 x - 5y=4x−5

Find a rectangular equation for the plane curve defined by the parametric equations. - x=5cos⁡t,y=−2sin⁡t;0≤t≤2πx = 5 \cos t , y = - 2 \sin t ; 0 \leq t \leq 2 \pix=5cost,y=−2sint;0≤t≤2π

Graph the curve whose parametric equations are given. - x=t+2,y=3t−1;0≤t≤3\mathrm{x}=\mathrm{t}+2, \mathrm{y}=3 \mathrm{t}-1 ; 0 \leq \mathrm{t} \leq 3x=t+2,y=3t−1;0≤t≤3

Find a rectangular equation for the plane curve defined by the parametric equations. - x=t3+1,y=t3−10;−2≤t≤2x = t ^ { 3 } + 1 , y = t ^ { 3 } - 10 ; - 2 \leq t \leq 2x=t3+1,y=t3−10;−2≤t≤2

Find a rectangular equation for the plane curve defined by the parametric equations. - x=3t,y=t+1;−2≤t≤3\mathrm { x } = 3 \mathrm { t } , \mathrm { y } = \mathrm { t } + 1 ; - 2 \leq \mathrm { t } \leq 3x=3t,y=t+1;−2≤t≤3

Graph the curve whose parametric equations are given. - x=2t2x=2 t^{2}x=2t2 , y=t+2y=t+2y=t+2 : −∞<t<∞- \infty < t < \infty−∞<t<∞

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 7 \sin t , y = 7 \cos t ; 0 \leq t \leq 2 \pi$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 2 t - 1 , y = t ^ { 2 } + 4 ; - 4 \leq t \leq 4$

Find parametric equations for the rectangular equation. - $y = x ^ { 4 } - 4$

Graph the curve whose parametric equations are given. - $x=3 \cos t, y=2 \sin t ; 0 \leq t \leq 2 \pi$ $Graph the curve whose parametric equations are given. - x=3 \cos t, y=2 \sin t ; 0 \leq t \leq 2 \pi A) B) C) D)$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 3 \tan t , y = 2 \sec t ; 0 \leq t \leq 2 \pi$

Find a rectangular equation for the plane curve defined by the parametric equations. - $y = 4 x - 5$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = 5 \cos t , y = - 2 \sin t ; 0 \leq t \leq 2 \pi$

Graph the curve whose parametric equations are given. - $\mathrm{x}=\mathrm{t}+2, \mathrm{y}=3 \mathrm{t}-1 ; 0 \leq \mathrm{t} \leq 3$ $Graph the curve whose parametric equations are given. - \mathrm{x}=\mathrm{t}+2, \mathrm{y}=3 \mathrm{t}-1 ; 0 \leq \mathrm{t} \leq 3 A) B) C) D)$

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = t ^ { 3 } + 1 , y = t ^ { 3 } - 10 ; - 2 \leq t \leq 2$

Find a rectangular equation for the plane curve defined by the parametric equations. - $\mathrm { x } = 3 \mathrm { t } , \mathrm { y } = \mathrm { t } + 1 ; - 2 \leq \mathrm { t } \leq 3$

Graph the curve whose parametric equations are given. - $x=2 t^{2}$ , $y=t+2$ : $- \infty < t < \infty$ $Graph the curve whose parametric equations are given. - x=2 t^{2} , y=t+2 : - \infty < t < \infty A) B) C) D)$